| L(s) = 1 | − 4·2-s − 8·3-s + 12·4-s + 10·5-s + 32·6-s − 20·7-s − 32·8-s + 24·9-s − 40·10-s + 44·11-s − 96·12-s − 26·13-s + 80·14-s − 80·15-s + 80·16-s − 92·17-s − 96·18-s − 132·19-s + 120·20-s + 160·21-s − 176·22-s − 336·23-s + 256·24-s + 75·25-s + 104·26-s − 88·27-s − 240·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.53·3-s + 3/2·4-s + 0.894·5-s + 2.17·6-s − 1.07·7-s − 1.41·8-s + 8/9·9-s − 1.26·10-s + 1.20·11-s − 2.30·12-s − 0.554·13-s + 1.52·14-s − 1.37·15-s + 5/4·16-s − 1.31·17-s − 1.25·18-s − 1.59·19-s + 1.34·20-s + 1.66·21-s − 1.70·22-s − 3.04·23-s + 2.17·24-s + 3/5·25-s + 0.784·26-s − 0.627·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 306 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 2396 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 92 T + 11822 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 132 T + 13004 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 336 T + 52288 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 18074 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 180 T + 17252 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 272 T + 110082 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 140 T + 142622 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 208 T + 114360 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1164 T + 545890 T^{2} + 1164 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 356 T + 286118 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 412 T + 170924 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 88 T + 374778 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 60 T + 602306 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 260 T + 185972 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1136 T + 857658 T^{2} - 1136 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 368 T + 602214 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 540 T + 409594 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 148 T + 1346294 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 452 T + 1852902 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.31087681153534602357963342930, −12.00880799337174171361483635295, −11.17492366804887707169916523967, −11.15978987969154511873133199784, −10.46916999060441166686741999858, −9.859906365925811825178062753975, −9.466587629121604375300082940657, −9.309191721533749509423186899588, −8.312018898016110558292456583047, −7.86659116024976856546571157642, −6.65198259477252451962445840132, −6.60128194743475326402726351835, −6.14135813364758669587683627909, −5.73823451377507164970640144488, −4.58298740369370365417808803169, −3.76271357158126718684963494125, −2.32077373382735662859775767465, −1.70455018970282064430612654844, 0, 0,
1.70455018970282064430612654844, 2.32077373382735662859775767465, 3.76271357158126718684963494125, 4.58298740369370365417808803169, 5.73823451377507164970640144488, 6.14135813364758669587683627909, 6.60128194743475326402726351835, 6.65198259477252451962445840132, 7.86659116024976856546571157642, 8.312018898016110558292456583047, 9.309191721533749509423186899588, 9.466587629121604375300082940657, 9.859906365925811825178062753975, 10.46916999060441166686741999858, 11.15978987969154511873133199784, 11.17492366804887707169916523967, 12.00880799337174171361483635295, 12.31087681153534602357963342930
